Three-Level Multi-Leader-Follower Incentive Stackelberg Differential Game with $H_{\mathrm{\infty }}$ Constraint

This paper is concerned with a three-level multi-leader-follower incentive Stackelberg game with $H_{\mathrm{\infty }}$ constraint. Based on $H_2\mathrm{/}{H}_{\mathrm{\infty }}$ control theory, we firstly obtain the worst-case disturbance and the team-optimal strategy by dealing with a nonzero-sum stochastic differential game. The main objective is to establish an incentive Stackelberg strategy set of the three-level hierarchy in which the whole system achieves the top leader’s team-optimal solution and attenuates the external disturbance under $H_{\mathrm{\infty }}$ constraint. On the other hand, followers on the bottom two levels in turn attain their state feedback Nash equilibrium, ensuring incentive Stackelberg strategies while considering the worst-case disturbance. By convex analysis theory, maximum principle and decoupling technique, the three-level incentive Stackelberg strategy set is obtained. Finally, a numerical example is given to illustrate the existence of the proposed strategy set.